Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 16 - Vector Calculus - 16.9 The Divergence Theorem - 16.9 Exercises - Page 1185: 6



Work Step by Step

Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $ where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}$ Here, $div F=\dfrac{\partial (x^2yz)}{\partial x}+\dfrac{\partial (xy^2z)}{\partial y}+\dfrac{\partial (xyz^2)}{\partial z}=6xyz$ Now, $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} =\int_{0}^a\int_0^b \int_0^c (6xyz) \times dzdydx$ $=\int_{0}^a\int_0^b [6xy \times (\dfrac{z^2}{2})]_0^cdydx$ $=\int_{0}^a\int_0^b (3xy) \cdot (c^2) dydx$ Hence, we have $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} =\dfrac{3}{4}a^2b^2c^2$
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